Here, \(T_{N_i}\)  denotes the decay temperature of the respective RHN species \(N_i\) , and \(T_{\Delta L = 2}\)  is the freeze-out temperature of the dimension-5 Weinberg operator in the seesaw extension of the Standard Model [1]}, with \(m_\nu \sim 0.05\,\textrm {GeV}\)  representing the mass scale of the light SM neutrinos.
Interactions mediated by the Weinberg operator violate total lepton number by two units, \(\Delta L = 2\) , and can thus play the role of the LNV interactions required for wash-in leptogenesis in the limit of very large RHN masses.
That is, for \(M_i \gtrsim 10^{13}\,\textrm {GeV}\) , LNV processes do not yet freeze out at the time of RHN decay, but only when lepton–Higgs scatterings with heavy off-shell RHNs in the intermediate state become inefficient at \(T \sim T_{\Delta L = 2}\) .\(17\endcsname \)Renormalizable interactions such as the Yukawa interactions of the SM fermions begin to enter thermal equilibrium as the temperature of the thermal bath steadily decreases.
The electron Yukawa interaction is, e.g., inefficient at temperatures above \(T_{y_e}\)  and efficient at temperatures below \(T_{y_e}\) .
By contrast, the interactions mediated by the Weinberg operator scale differently with temperature because of the dimensionful coupling constant and thus leave thermal equilibrium at temperatures around \(T_{\Delta L = 2}\) .

The estimate in Eq. (REF ) tells us that we can easily realize wash-in leptogenesis at higher temperatures simply by increasing the RHN masses \(M_i\) .
Wash-in leptogenesis can in particular occur in the temperature regimes (iv), (iii) and (ii), which we introduced in Sec. REF , if we choose the lightest RHN mass, \(M_1\) , to lie in one of the intervals \(\left(10^6,10^9\right)\,\textrm {GeV}\) , \(\left(10^9,10^{11\cdots 12}\right)\,\textrm {GeV}\) , and \(\left(10^{11\cdots 12},10^{13}\right)\,\textrm {GeV}\) , respectively.
For completeness, we will also consider wash-in leptogenesis in temperature regime (i), \(T \in \left(10^{13},10^{15}\right)\,\textrm {GeV}\) , in the following.
In this way, we will be able to account for the uncertainty in the freeze-out temperature of the Weinberg operator \(T_{\Delta L = 2}\) , which, as can be seen from Eq. (REF ), may not be too different from the lower boundary of temperature regime (i).
